Point P is on line segment start overline, O, Q, end overline 
OQ
​
 . Given O, Q, equals, 4, x, minus, 10, commaOQ=4x−10, O, P, equals, 2, x, commaOP=2x, and P, Q, equals, x, commaPQ=x, determine the numerical length of start overline, O, P, end overline, . 
OP
 .

We are given three lengths for the line segment \( \overline{OQ} \) and the two parts into which point \( P \) divides it:

- \( OQ = 4x - 10 \)
- \( OP = 2x \)
- \( PQ = x \)

Since point \( P \) is on line segment \( \overline{OQ} \), the sum of the lengths of \( OP \) and \( PQ \) must equal the total length \( OQ \). Therefore, we can write the equation:

\[
OP + PQ = OQ
\]

Substitute the given values into the equation:

\[
2x + x = 4x - 10
\]

Simplify the left-hand side:

\[
3x = 4x - 10
\]

Now, subtract \( 4x \) from both sides:

\[
3x - 4x = -10
\]

Simplify further:

\[
-x = -10
\]

Multiply both sides by \( -1 \):

\[
x = 10
\]

Now that we know \( x = 10 \), we can substitute this value into the expression for \( OP \):

\[
OP = 2x = 2(10) = 20
\]

Thus, the numerical length of \( OP \) is \( 20 \).

Would you like more details or have any questions about this? Here are 5 related questions you might explore:

1. What is the length of \( PQ \) using the value of \( x \)?
2. How would the result change if \( PQ \) were doubled?
3. What would happen if \( OP \) were half of its original length?
4. How can we confirm the total length of \( OQ \)?
5. Can we generalize the result for any similar division of a line segment?

**Tip:** Always check that the sum of the parts matches the total length in geometry problems involving segments.

Point P is on line segment OQ. Given OQ = 4x - 10, OP = 2x, and PQ = x, determine the numerical length of OP.

Find the length of segment OP in a line divided by point P using the algebraic expressions OQ = 4x - 10, OP = 2x, and PQ = x. This problem involves basic algebra and the Segment Addition Postulate, making it suitable for Grades 6-8.

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